I have a few, I'm an old fart so some of you won't of heard these.
Grace by Jeff Buckley. Lovely song and Matt wouldn't naff it up!
Blue Velvet.....I can't remember the artist that did this one, but I remember it was in a film with Izzabella Rossillinni in. Very haunting melody. Would suit Matt's voice
People are Strange by Echo and the Bunnymen, just for the fun value on that.
Electricity by O.M.D Replacing almost all the synths with guitar!
Entangled by Genesis. If you have ever heard THIS one you'll know why I picked it!